The anisotropic and region-dependent mechanical response of wrap-around tendons under tensile, compressive and combined multiaxial loads.

The present work reports on the multiaxial region and orientation-dependent mechanical properties of two porcine wrap-around tendons under tensile, compressive and combined loads based on an extensive study with n=175 samples. The results provide a detailed dataset of the anisotropic tensile and compressive longitudinal properties and document a pronounced tension-compression asymmetry. Motivated by the physiological loading conditions of these tendons, which include transversal compression at bony abutments in addition to longitudinal tension, we systematically investigated the change in axial tension when the tendon is compressed transversally along one or both perpendicular directions. The results reveal that the transversal compression can increase axial tension (proximal-distal direction) in both cases to orders of 30%, yet by a larger amount in the first case (transversal compression in anterior-posterior direction), which seems to be more relevant for wrap-around tendons in-vivo. These quantitative measurements are in line with earlier findings on auxetic properties of tendon tissue, but show for the first time the influence of this property on the stress response of the tendon, and may thus reveal an important functional principle within these essential elements of force transmission in the body. STATEMENT OF SIGNIFICANCE: The work reports for the first time on multiaxial region and orientation-dependent mechanical properties of wrap-around tendons under various loads. The results indicate that differences in the mechanical properties exist between zones that are predominantly in a uniaxial tensile state and those that experience complex load states. The observed counterintuitive increase of the axial tension upon lateral compression points at auxetic properties of the tendon tissue which may be pivotal for the function of the tendon as an element of the musculoskeletal system. It suggests that the tendon's performance in transmitting forces is not diminished but enhanced when the action line is deflected by a bony pulley around which the tendon wraps, representing an important functional principle of tendon tissue.

On multiscale tension-compression asymmetry in skeletal muscle.

Skeletal muscle tissue shows a clear asymmetry with regard to the passive stresses under tensile and compressive deformation, referred to as tension-compression asymmetry (TCA). The present study is the first one reporting on TCA at different length scales, associated with muscle tissue and muscle fibres, respectively. This allows for the first time the comparison of TCA between the tissue and one of its individual components, and thus to identify the length scale at which this phenomenon originates. Not only the passive stress-stretch characteristics were recorded, but also the volume changes during the axial tension and compression experiments. The study reveals clear differences in the characteristics of TCA between fibres and tissue. At tissue level TCA increases non-linearly with increasing deformation and the ratio of tensile to compressive stresses at the same magnitude of strain reaches a value of approximately 130 at 13.5% deformation. At fibre level instead it initially drops to a value of 6 and then rises again to a TCA of 14. At a deformation of 13.5%, the tensile stress is about 6 times higher. Thus, TCA is about 22 times more expressed at tissue than fibre scale. Moreover, the analysis of volume changes revealed little compressibility at tissue scale whereas at fibre level, especially under compressive stress, the volume decreases significantly. The data collected in this study suggests that the extracellular matrix has a distinct role in amplifying the TCA, and leads to more incompressible tissue behaviour. STATEMENT OF SIGNIFICANCE: This article analyses and compares for the first time the tension-compression asymmetry (TCA) displayed by skeletal muscle at tissue and fibre scale. In addition, the volume changes of tissue and fibre specimens with application of passive tensile and compressive loads are studied. The study identifies a key role of the extracellular matrix in establishing the mechanical response of skeletal muscle tissue: It contributes significantly to the passive stress, it is responsible for the major part of tissue-scale TCA and, most probably, prevents/balances the volume changes of muscle fibres during deformation. These new results thus shed light on the origin of TCA and provide new information to be used in microstructure-based approaches to model and simulate skeletal muscle tissue.

Age-dependent mechanical and microstructural properties of the rabbit soleus muscle.

During growth there are serious changes in the skeletal muscles to compensate for the changed requirements in terms of body weight and size. In this study, the age-dependent (between 21 and 100 days) mechanical and microstructural properties of rabbit soleus muscle tissue were investigated. For this purpose, morphological properties (animal mass, soleus muscle mass, tibial length) were measured at 5 different times during aging. On the other hand, fibre orientation-dependent axial and semi-confined compression experiments were realised. In addition, the essential components (muscle fibres, extracellular matrix, remaining components), dominating the microstructure of muscle tissue, were analysed. While the mechanical results show hardly any age-dependent differences, the morphological and microstructural results show clear age-dependent differences. All morphological parameters increase significantly (animal mass by 839.2%, muscle mass 1050.6%, tibial length 233.6%). In contrast, microstructural parameters change differently. The percentage of fibres (divided into slow-twitch (ST) and fast-twitch (FT) fibres) increases significantly (137.6%), while the proportions of the extracellular matrix and the remaining components (48.2% and 46.1%) decrease. At the same time, the cross-sectional area of the fibres increases significantly (697.9%). The totality of this age-dependent information provides a deeper understanding of age-related changes in muscle structure and function and may contribute to successful development and validation of growth models in the future. STATEMENT OF SIGNIFICANCE: This article reports the first comprehensive data set on age-dependent morphological (animal mass, soleus muscle mass, tibial length), mechanical (axial and semi-confined compression), and microstructural (muscle fibres, extracellular matrix, remaining components) properties of the rabbit soleus muscle. On the one hand, the results of this study contribute to the understanding of muscle mechanics and thus to understanding of load transfer mechanisms inside the muscle tissue during growth. On the other hand, these results are relevant to the fields of constitutive formulation of age-dependent muscle tissue.

On a phase-field approach to model fracture of small intestine walls.

We address anisotropic elasticity and fracture in small intestine walls (SIWs) with both experimental and computational methods. Uniaxial tension experiments are performed on porcine SIW samples with varying alignments and quantify their nonlinear elastic anisotropic behavior. Fracture experiments on notched SIW strips reveal a high sensitivity of the crack propagation direction and the failure stress on the tissue orientation. From a modeling point of view, the observed anisotropic elastic response is studied with a continuum mechanical model stemming from a strain energy density with a neo-Hookean component and an anisotropic component with four families of fibers. Fracture is addressed with the phase-field approach, featuring two-fold anisotropy in the fracture toughness. Elastic and fracture model parameters are calibrated based on the experimental data, using the maximum and minimum limits of the experimental stress-stretch data set. A very good agreement between experimental data and computational results is obtained, the role of anisotropy being effectively captured by the proposed model in both the elastic and the fracture behavior. STATEMENT OF SIGNIFICANCE: This article reports a comprehensive experimental data set on the mechanical failure behavior of small intestinal tissue, and presents the corresponding protocols for preparing and testing the samples. On the one hand, the results of this study contribute to the understanding of small intestine mechanics and thus to understanding of load transfer mechanisms inside the tissue. On the other hand, these results are used as input for a phase-field modelling approach, presented in this article. The presented model can reproduce the mechanical failure behavior of the small intestine in an excellent way and is thus a promising tool for the future mechanical description of diseased small intestinal tissue.

Predicting muscle tissue response from calibrated component models and histology-based finite element models.

Skeletal muscle is an anisotropic soft biological tissue composed of muscle fibres embedded in a structurally complex, hierarchically organised extracellular matrix. In a recent work (Kuravi et al., 2021) we have developed 3D finite element models from series of histological sections. Moreover, based on decellularisation of fresh tissue samples, a novel set of experimental data on the direction dependent mechanical properties of collagenous ECM was established (Kohn et al., 2021). Together with existing information on the material properties of single muscle fibres, the combination of these techniques allows computing predictions of the composite tissue response. To this end, an inverse finite element procedure is proposed in the present work to calibrate a constitutive model of the extracellular matrix, and supplementary biaxial tensile tests on fresh and decellularised tissues are performed for model validation. The results of this rigorously predictive and thus unforgiving strategy suggest that the prediction of the tissue response from the individual characteristics of muscle cells and decellularised tissue is only possible within clear limits. While orders of magnitude are well matched, and the qualitative behaviour in a wide range of load cases is largely captured, the existing deviations point at potentially missing components of the model and highlight the incomplete experimental information in bottom-up multiscale approaches to model skeletal muscle tissue.

Direct measurement of the direction-dependent mechanical behaviour of skeletal muscle extracellular matrix.

This paper reports the first comprehensive data set on the anisotropic mechanical properties of isolated endo- and perimysial extracellular matrix of skeletal muscle, and presents the corresponding protocols for preparing and testing the samples. In particular, decellularisation of porcine skeletal muscle is achieved with caustic soda solution, and mechanical parameters are defined based on compressive and tensile testing in order to identify the optimal treatment time such that muscle fibres are dissolved whereas the extracellular matrix remains largely intact and mechanically functional. At around 18 h, a time window was found and confirmed by histology, in which axial tensile experiments were performed to characterise the direction-dependent mechanical response of the extracellular matrix samples, and the effect of lateral pre-compression was studied. The typical, large variability in the experimental stress response could be largely reduced by varying a single scalar factor, which was attributed to the variation of the fraction of extracellular matrix within the tissue. While experimental results on the mechanical properties of intact muscle tissue and single muscle fibres are increasingly available in literature, there is a lack of information on the properties of the collagenous components of skeletal muscle. The present work aims at closing this gap and thus contributes to an improved understanding of the mechanics of skeletal muscle tissue and provides a missing piece of information for the development of corresponding constitutive and computational models.

Exhaustion of Skeletal Muscle Fibers Within Seconds: Incorporating Phosphate Kinetics Into a Hill-Type Model.

Initiated by neural impulses and subsequent calcium release, skeletal muscle fibers contract (actively generate force) as a result of repetitive power strokes of acto-myosin cross-bridges. The energy required for performing these cross-bridge cycles is provided by the hydrolysis of adenosine triphosphate (ATP). The reaction products, adenosine diphosphate (ADP) and inorganic phosphate (P i ), are then used-among other reactants, such as creatine phosphate-to refuel the ATP energy storage. However, similar to yeasts that perish at the hands of their own waste, the hydrolysis reaction products diminish the chemical potential of ATP and thus inhibit the muscle's force generation as their concentration rises. We suggest to use the term "exhaustion" for force reduction (fatigue) that is caused by combined P i and ADP accumulation along with a possible reduction in ATP concentration. On the basis of bio-chemical kinetics, we present a model of muscle fiber exhaustion based on hydrolytic ATP-ADP-P i dynamics, which are assumed to be length- and calcium activity-dependent. Written in terms of differential-algebraic equations, the new sub-model allows to enhance existing Hill-type excitation-contraction models in a straightforward way. Measured time courses of force decay during isometric contractions of rabbit M. gastrocnemius and M. plantaris were employed for model verification, with the finding that our suggested model enhancement proved eminently promising. We discuss implications of our model approach for enhancing muscle models in general, as well as a few aspects regarding the significance of phosphate kinetics as one contributor to muscle fatigue.

Changes in three-dimensional muscle structure of rabbit gastrocnemius, flexor digitorum longus, and tibialis anterior during growth.

Muscular contraction dynamics depends on active and passive muscle properties (e.g., the force-velocity relation) as well as on the three-dimensional (3D) muscle structure (e.g., the muscle fascicle architecture and aponeurosis dimensions). Much is known about active muscle force generation and the muscle architecture at a particular age (mostly for adult specimens), but less is known about changes in muscle structure during growth. The present study analyzed growth-related changes in the muscle structure of rabbit gastrocnemius lateralis (GL), gastrocnemius medialis (GM), flexor digitorum longus (FDL), and tibialis anterior (TA). Changes in tendon length, muscle belly dimensions (length, width, thickness), as well as aponeurosis length, width, and area were determined using 55 rabbits between 18 and 108 days old. Additionally, the 3D muscle fascicle architecture of five rabbits of different ages (21, 37, 50, 70, 100 days) was determined using a manual digitizer. We found an almost linear increase over time in most of the geometrical parameters observed. GL and GM showed very similar growth characteristics. In contrast to the pronounced increase in muscle belly length of GL and GM, FDL and TA exhibited more uniform muscle belly growth in length, width, and thickness. In general, the aponeuroses of the muscles exhibited lower growth rates in width than in length, and aponeurosis areas were larger than physiological cross-sectional areas. There were almost no changes in fascicle lengths with increasing age for GL, GM, and FDL. In contrast, there was a clear increase in TA fascicle length from about 20 to over 40mm. Pennation angles of TA (11.0 ± 2.1°) and FDL (16.7 ± 3.2°) remained almost unchanged but increased for GL from 13.4 ± 3.3° to 24.3 ± 6.5° from the youngest to the oldest animal. For all muscles observed, the tendon-muscle fascicle length ratio (rTFL) changed during growth. GL and GM exhibited similar increases in rTFL from about 4-8. FDL showed the highest ratio, which increased from about 8-13, whereas TA had the lowest ratio, which decreased slightly from 2 to 1.5. The outcomes demonstrate new findings regarding changes in 3D muscle architecture and aponeurosis shape during growth, and they provide information for muscle force generation, functional relevance, and adaptation with respect to animal age. Therefore, the results help to improve understanding of muscle growth processes and can be used as input data for muscle growth modeling.

Effects of Growth on Muscle, Tendon, and Aponeurosis Tissues in Rabbit Shank Musculature.

There exist several studies using morphological analyses of skeletal muscles to obtain a better understanding of muscle structure. The structural information obtained are primarily determined from single muscle components using individual animals of discrete ages. Further, little is known about changing dimensions of the aponeurosis, which is an important load-transferring interface in muscle mechanics. Thus, the aim of the present study was to determine how the muscle, tendon, and particularly the aponeurosis geometry of the rabbit shank musculature (M. soleus, M. extensor digitorum longus, and M. plantaris) change during growth. In doing so, morphological studies on muscles of eighty-nine female rabbits aged between 18 and 108 days were conducted. We found an almost linear increase over time in all of the geometrical parameters observed. The aponeurosis of the muscles exhibited lower growth rates in width than in length. The distal and proximal aponeurosis areas were nearly identical. The ratio of aponeurosis area to the physiological cross-sectional area was 2.54, 2.54, and 1.88 for M. soleus, M. extensor digitorum longus, and M. plantaris, respectively. M. extensor digitorum longus and M. soleus exhibited a nearly similar tendon-muscle fascicle length ratio during growth, increasing from 2.86 to 5.30 and 3.48 to 6.16, respectively. Interestingly, the tendon-muscle fascicle length ratio of the M. plantaris started initially with a much higher value (∼8) and increased to ∼18. Taken together, these results provide insight into the structure of the muscle-tendon complex and thus, a general understanding of muscle growth. Anat Rec, 300:1123-1136, 2017. © 2016 Wiley Periodicals, Inc.

Intermuscular pressure between synergistic muscles correlates with muscle force.

The purpose of the study was to examine the relationship between muscle force generated during isometric contractions (i.e. at a constant muscle-tendon unit length) and the intermuscular (between adjacent muscles) pressure in synergistic muscles. Therefore, the pressure at the contact area of the gastrocnemius and plantaris muscle was measured synchronously to the force of the whole calf musculature in the rabbit species Oryctolagus cuniculus Similar results were obtained when using a conductive pressure sensor, or a fibre-optic pressure transducer connected to a water-filled balloon. Both methods revealed a strong linear relationship between force and pressure in the ascending limb of the force-length relationship. The shape of the measured force-time and pressure-time traces was almost identical for each contraction (r=0.97). Intermuscular pressure ranged between 100 and 700 mbar (70,000 Pa) for forces up to 287 N. These pressures are similar to previous (intramuscular) recordings within skeletal muscles of different vertebrate species. Furthermore, our results suggest that the rise in intermuscular pressure during contraction may reduce the force production in muscle packages (compartments).

Long-term mechanical behaviour of skeletal muscle tissue in semi-confined compression experiments.

In this study, porcine skeletal muscle tissue was tested until 112 hours post mortem using a semi-confined compression device that induces fascicles to enter one of the states of compression (mode I), tension (mode II), or constant length (mode III). Based on the authors׳ previous studies (Böl et al., 2014, 2015a), the anisotropic mechanical behaviour of the tissue was analysed, with a special focus on the testing time post mortem. The results suggest that the tissue exhibits significant anisotropic behaviour during the first hours of post mortem but that this anisotropy becomes insignificant at later time points. Interestingly, the compressibility of the tissue is more or less unaffected by the testing time. These results are discussed especially with respect to tissue microstructure.

Novel microstructural findings in M. plantaris and their impact during active and passive loading at the macro level.

There are several studies dealing with experimental and structural analyses of skeletal muscles that are aimed at gaining a better understanding of three-dimensional muscle deformation and force generation. A variety of these contributions have performed structural or mechanical analyses, but very few have combined these approaches at different levels. To fill this gap, the present study aims to bring together three-dimensional micro-structural and mechanical findings in rabbit M. plantaris to study load transfer mechanisms inside the muscle during passive loading and active muscle contraction. During these two deformation states, the three-dimensional surface of the aponeurosis-tendon complex was recorded using optical measurement systems. In this way, the strain distribution on the muscle can be calculated to interpret the load transfer mechanisms inside the muscle. The results show that the three-dimensional strain distribution during muscle activation is completely different from the distribution during passive loading. Under both loading conditions, the strain distribution is irregular. To interpret these findings, the gross try and the fascicle architecture of the M. plantaris were determined. In doing so, a highly complex microstructure featuring tube- and sail-like structure was identified. Moreover, a compartmentalisation of the muscle into two compartments was detected. The smaller, bipennated muscle compartment was embedded into the larger, unipennated compartment. To the authors' knowledge, this type of inner structure has never been previously documented in single-headed muscles.

Three-Dimensional Muscle Architecture and Comprehensive Dynamic Properties of Rabbit Gastrocnemius, Plantaris and Soleus: Input for Simulation Studies.

The vastly increasing number of neuro-muscular simulation studies (with increasing numbers of muscles used per simulation) is in sharp contrast to a narrow database of necessary muscle parameters. Simulation results depend heavily on rough parameter estimates often obtained by scaling of one muscle parameter set. However, in vivo muscles differ in their individual properties and architecture. Here we provide a comprehensive dataset of dynamic (n = 6 per muscle) and geometric (three-dimensional architecture, n = 3 per muscle) muscle properties of the rabbit calf muscles gastrocnemius, plantaris, and soleus. For completeness we provide the dynamic muscle properties for further important shank muscles (flexor digitorum longus, extensor digitorum longus, and tibialis anterior; n = 1 per muscle). Maximum shortening velocity (normalized to optimal fiber length) of the gastrocnemius is about twice that of soleus, while plantaris showed an intermediate value. The force-velocity relation is similar for gastrocnemius and plantaris but is much more bent for the soleus. Although the muscles vary greatly in their three-dimensional architecture their mean pennation angle and normalized force-length relationships are almost similar. Forces of the muscles were enhanced in the isometric phase following stretching and were depressed following shortening compared to the corresponding isometric forces. While the enhancement was independent of the ramp velocity, the depression was inversely related to the ramp velocity. The lowest effect strength for soleus supports the idea that these effects adapt to muscle function. The careful acquisition of typical dynamical parameters (e.g. force-length and force-velocity relations, force elongation relations of passive components), enhancement and depression effects, and 3D muscle architecture of calf muscles provides valuable comprehensive datasets for e.g. simulations with neuro-muscular models, development of more realistic muscle models, or simulation of muscle packages.

Tissue-scale anisotropy and compressibility of tendon in semi-confined compression tests.

In this study, porcine tendon tissue was tested with a dedicated semi-confined compression set-up that enables us to induce states of either fibrils in compression (mode I), tension (mode II) or at constant length (mode III), respectively. The results suggest that tendon tissue is compressible and demonstrates a significantly stiffer response in mode I than in mode III. This implies that the fibril direction remains the axis of transverse isotropy in compression and that it provides an anisotropic contribution to the tissue stress. These results, which are important for the development of constitutive models for tendon tissue, are discussed with respect to the hierarchical structure of the extracellular matrix.

On the anisotropy of skeletal muscle tissue under compression.

This paper deals with the role of the muscle fibres and extracellular matrix (ECM) components when muscle tissue is subjected to compressive loads. To this end, dissected tissue samples were tested in compression modes which induced states of fibres in compression (I), in tension (II) or at constant length (III), respectively. A comparison of the stress responses indicated that the tissue behaviour is significantly different for these modes, including differences between the modes (I) and (III). This contradicts the paradigm of many constitutive models that the stress response can be decomposed into an isotropic part relating to the ECM and an anisotropic fibre part the contribution of which can be neglected under compression. Conversely, the results provide experimental evidence that there is an anisotropic contribution of the fibre direction to the compressive stress. Interpreting these results in terms of recent microscopical studies, potential connections between the observed behaviour and the structure of muscle ECM are established.

Three-dimensional surface geometries of the rabbit soleus muscle during contraction: input for biomechanical modelling and its validation.

There exists several numerical approaches to describe the active contractile behaviour of skeletal muscles. These models range from simple one-dimensional to more advanced three-dimensional ones; especially, three-dimensional models take up the cause of describing complex contraction modes in a realistic way. However, the validation of such concepts is challenging, as the combination of geometry, material and force characteristics is so far not available from the same muscle. To this end, we present in this study a comprehensive data set of the rabbit soleus muscle consisting of the muscles' characteristic force responses (active and passive), its three-dimensional shape during isometric, isotonic and isokinetic contraction experiments including the spatial arrangement of muscle tissue and aponeurosis-tendon complex, and the fascicle orientation throughout the whole muscle at its optimal length. In this way, an extensive data set is available giving insight into the three-dimensional geometry of the rabbit soleus muscle and, further, allowing to validate three-dimensional numerical models.

Compressive properties of passive skeletal muscle-the impact of precise sample geometry on parameter identification in inverse finite element analysis.

Due to the increasing developments in modelling of biological material, adequate parameter identification techniques are urgently needed. The majority of recent contributions on passive muscle tissue identify material parameters solely by comparing characteristic, compressive stress-stretch curves from experiments and simulation. In doing so, different assumptions concerning e.g. the sample geometry or the degree of friction between the sample and the platens are required. In most cases these assumptions are grossly simplified leading to incorrect material parameters. In order to overcome such oversimplifications, in this paper a more reliable parameter identification technique is presented: we use the inverse finite element method (iFEM) to identify the optimal parameter set by comparison of the compressive stress-stretch response including the realistic geometries of the samples and the presence of friction at the compressed sample faces. Moreover, we judge the quality of the parameter identification by comparing the simulated and experimental deformed shapes of the samples. Besides this, the study includes a comprehensive set of compressive stress-stretch data on rabbit soleus muscle and the determination of static friction coefficients between muscle and PTFE.